Human contribution summary on Math-2504: Scott Duke Kominers

Human Contribution: Scott Duke Kominers

Multi-covering extension of OpenAI Result 092

Paper and attribution.

Scott Duke Kominers, Multi-coverings by lattice translates of convex bodies, hexagon:2610.00165v1, is a preliminary manuscript dated October 8, 2026. It develops the single-lattice construction in OpenAI's Result 092 into a bound for arbitrary covering multiplicity. The author credits substantial LLM assistance, principally GPT-6 Astra, while taking responsibility for the problem, methods, exposition, and any errors. This record is a summary of that authored contribution.

Lattice multi-coverings (Theorem 1.1).

For a convex body $K\subset\mathbb R^n$ and a full-rank lattice $\Lambda$, write \[ N_{\Lambda,K}(x)=\sum_{\lambda\in\Lambda}\mathbf 1_K(x-\lambda), \qquad \rho=\frac{\operatorname{vol}_n(K)}{\det\Lambda}. \] A lattice $k$-fold covering requires $N_{\Lambda,K}(x)\geq k$ at every $x$, with one translate for each distinct lattice point. Its density $\rho$ is the mean multiplicity. Let $\theta_{\mathrm{lat},k}(K)$ be the infimum of these densities. Theorem 1.1 states that, for every $\varepsilon>0$, a constant $C_\varepsilon$ depending only on $\varepsilon$ gives \[ \theta_{\mathrm{lat},k}(K) \leq \max\{(1+\varepsilon)k,\ C_\varepsilon n\log n\} \quad(n\geq2,\ k\geq1). \] No symmetry assumption is imposed on $K$. In particular, the density approaches the unavoidable lower bound $k$ uniformly over bodies and dimensions when $k/(n\log n)$ tends to infinity.

Prescribed density and simultaneous coverings (Theorem 1.2).

For one body and $0<\varepsilon<1$, every prescribed density $\rho\geq A_\varepsilon n\log n$ can be realized by a lattice satisfying $N_{\Lambda,K}(x)\geq(1-\varepsilon)\rho$ everywhere, in every dimension $n\geq2$. In sufficiently large dimension, the theorem also constructs one lattice of a prescribed covolume $\Delta$ for up to $\exp(n^{c_0})$ bodies, for an absolute $c_0>0$. This simultaneous statement assumes that their uniform distributions share a covariance matrix and that each mean $\rho_j=\operatorname{vol}_n(K_j)/\Delta$ exceeds the threshold. The bodies can have different volumes; their means need not agree. There is no matching pointwise upper bound on multiplicity or claim about a typical random lattice.

Worst-case order (Corollary 1.3).

Combining Theorem 1.1 with Li--Liu's earlier one-fold lattice lower bound, the paper obtains worst-case density of order $k+n\log n$, both over all convex bodies and over origin-symmetric convex bodies. This is a worst-case lattice statement; it does not identify the optimal one-fold translative density for a particular body such as a ball.

Outline of the argument.

The proof adapts OpenAI's horizontal--vertical decomposition of a lattice. Approximately Gaussian section volumes provide layer weights, and averaging a bounded terminal block over its cosets retains almost all of the mean while keeping individual selected weights small. Independent spatial thinning turns a mean-hole estimate into a Laplace-transform bound for section counts. A random shear combines those estimates to make the region of insufficient multiplicity very small. A small dilation eliminates that region; a final lattice rescaling gives the prescribed mean. Equal covariance permits a common normalization in the simultaneous version.

Scope relative to this problem.

The recorded problem asks for the optimal asymptotic density of a one-fold covering by congruent Euclidean balls. Specializing Theorem 1.1 to a ball gives a lattice $k$-fold covering with the stated density bound and, at $k=1$, an $O(n\log n)$ upper bound. For multiplicities large compared with $n\log n$, it gives density asymptotic to $k$. These conclusions are partial progress relative to the recorded one-fold question: they do not determine its optimal asymptotic density. Corollary 1.3 concerns worst-case convex bodies under the lattice restriction and supplies no matching one-fold lower bound for balls.

Status and provenance.

This site records the paper's reported results and has not independently reviewed the proof. The contribution is displayed in the Human Contribution collection, with its disclosed machine assistance; it is separate from the OpenAI claim and from any independent community review.

Sources.

• Scott Duke Kominers, Multi-coverings by lattice translates of convex bodies, version 1: Theorems 1.1 and 1.2, Corollary 1.3, introduction, and author assistance statement. Paper and provenance; PDF.
• OpenAI, A single-lattice covering bound of order $n\log n$, Result 092, manuscript dated September 23, 2026. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026/paper.pdf.
• H. Li and X. Liu, Nearly sharp bounds for lattice coverings by convex bodies, version 2; the lower bound used in Corollary 1.3. https://arxiv.org/abs/2607.28429v2.